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Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indis- pensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which ref- erence is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I ar- gue that these two arguments are in conflict with each other. Whereas the indispensability argument supports realism about mathematics, the indeter- minacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics.

After reviewing some different indispensability arguments, I distinguish several different ways in which mathematics can make an important contribution to a scientific explanation. Once these contributions are highlighted it will be possible to see that indispensability arguments have little chance of convincing us of the existence of abstract objects, even though they may give us good reason to accept the truth of some mathematical claims. However, in the concluding part of this paper, I argue that even though there is a valid indispensability argument for realism about some mathematical claims, this argument is problematic as it begs the question at issue. This challenge to indispensability arguments is then used to suggest that if mathematics is making these sorts of contributions to science, then it may be the case that mathematical claims receive some non-empirical support prior to their application in scientific explanation.

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine’s “Indispensability Argument”, which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics . I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field’s nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field’s fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer’s argument for his thesis that fictionalism is “the best version of anti-realistic anti-platonism”.

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage.

In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters.

Quine and Putnam's Indispensability Argument claims that we must be ontologically committed to mathematical objects, because of the indispensability of mathematics in our best scientific theories. Indispensability means that physical theories refer to and quantify over mathematical entities such as sets, numbers and functions. In his famous book 'Science Without Numbers' Hartry Field argues that this is not the case. We can "nominalize" our physical theories, that is we can reformulate them in such a way that 1) the new version preserves the attractivity of the theory, and 2) the nominalized theory does not contain quantifications over mathematical entities. I'm going to reconsider Field's nominalization procedure for a toy physic theory formulated in a first order language, in order to make a clear distinction between the following three steps: -the physical theory in terms of empirical observations. -the standard physical theory, which contains quantification over mathematical entities, as usual. -the nominalized version of the theory without any reference to mathematical entities. Having Field's nominalization procedure reconstructed, it will be clear that there is no difference between the original and the nominalized versions of the theory, at least, there is no difference from a formalist point of view. It is because the only difference would come from the different "meanings" of the variables over which the quantifications are running. The formalist philosophy of mathematics, however, denies that the variables have meanings at all. So, the formal systems as abstract mathematical entities are still included to physical theories; and this fact is highly enough for the structural platonist or immanent realist to apply the Quine-Putnam argument. Finally, therefore, I will suggest a completely different way for the objection to the Quine-Putnam argument.

One of the most influential arguments for realism about mathematical objects is the indispensability argument. Simply put, this is the argument that insofar as we are committed to the existence of the physical objects existentially quantified over in our best scientific theories, we are also committed to the mathematical objects existentially quantified over in these theories. Following the Quine–Putnam formulation of the indispensability argument, some proponents of the indispensability argument have made the mistake of taking confirmational holism to be an essential premise of the argument. In this paper, I consider the reasons philosophers have taken confirmational holism to be essential to the argument and argue that, contrary to the traditional view, confirmational holism is dispensable.

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists..

This paper considers two strategies for undermining indispensability arguments for mathematical Platonism. We defend one strategy (the Trivial Strategy) against a criticism by Joseph Melia. In particular, we argue that the key example Melia uses against the Trivial Strategy fails. We then criticize Melia’s chosen strategy (the Weaseling Strategy.) The Weaseling Strategy attempts to show that it is not always inconsistent or irrational knowingly to assert p and deny an implication of p . We argue that Melia’s case for this strategy fails.